Mobility in Symmetry-Regular Bar-and-Joint Frameworks

  • P. W. Fowler
  • S. D. GuestEmail author
  • B. Schulze
Part of the Fields Institute Communications book series (FIC, volume 70)


In a symmetry-regular bar-and-joint framework of given point-group symmetry, all bars and joints occupy general positions with respect to the symmetry elements. The symmetry-extended form of Maxwell’s Rule is applied to this simplest type of framework and is used to derive counts within irreducible representations for infinitesimal mechanisms and states of self stress. In particular, conditions are given for symmetry-regular frameworks to have at least one infinitesimal mechanism (respectively, state of self stress) within each irreducible representation of the point group of the framework. Similar conditions are found for symmetry-regular body-and-joint frameworks.


Infinitesimal rigidity Mechanisms Frameworks Symmetry 

Subject Classifications

52C25 70B15 20C35 



This work was begun during the workshop on Rigidity and Symmetry, held at the Fields Institute, Toronto, October 17–21, 2011, and gained from discussions there with Prof. Stephen Power (Lancaster). PWF thanks the Fields Institute for financial support of his visit.


  1. 1.
    Altmann, S.L., Herzig, P.: Point-Group Theory Tables. Clarendon Press, Oxford (1994)Google Scholar
  2. 2.
    Atkins, P.W., Child, M.S., Phillips, C.S.G.: Tables for Group Theory. Oxford University Press, London (1970)Google Scholar
  3. 3.
    Bishop, D.M.: Group Theory and Chemistry. Clarendon Press, Oxford (1973)Google Scholar
  4. 4.
    Bottema, O.: Die Bahnkurven eines merkwürdigen Zwölfstabgetriebes. Österr. Ingen. Archiv 14, 218–222 (1960)zbMATHGoogle Scholar
  5. 5.
    Connelly, R., Fowler, P.W., Guest, S.D., Schulze, B., Whiteley, W.J.: When is a symmetric pin-jointed framework isostatic? Int. J. Solids Struct. 46, 762–773 (2009)CrossRefzbMATHGoogle Scholar
  6. 6.
    Fowler, P.W., Guest, S.D.: A symmetry extension of Maxwell’s rule for rigidity of frames. Int. J. Solids Struct. 37, 1793–1804 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Fowler, P.W., Guest, S.D.: Symmetry analysis of the double banana and related indeterminate structure. In: Drew, H.R., Pellegrino, S. (eds.) New Approaches to Structural Mechanics, Shells and Biological Structures, pp. 91–100. Kluwer Academic, Dordrecht (2002)CrossRefGoogle Scholar
  8. 8.
    Fowler, P.W., Guest, S.D.: Symmetry and states of self stress in triangulated toroidal frames. Int. J. Solids Struct. 39(17), 4385–4393 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Fowler, P.W., Guest, S.D.: A symmetry analysis of mechanisms in rotating rings of tetrahedra. Proc. R. Soc.: Math. Phys. Eng. Sci. 461, 1829–1846 (2005)Google Scholar
  10. 10.
    Fowler, P.W., Guest, S.D., Tarnai, T.: A symmetry treatment of Danzerian rigidity for circle packing. Proc. R. Soc. A: Math. Phys. Eng. Sci. 464, 3237–3254 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Guest, S.D.: Tensegrities and rotating rings of tetrahedra: a symmetry viewpoint of structural mechanics. Philos. Trans. R. Soc. Lond. 358(358), 229–243 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Guest, S.D., Fowler, P.W.: A symmetry-extended mobility rule. Mech. Mach. Theory 40, 1002–1014 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Guest, S.D., Fowler, P.W.: Symmetry conditions and finite mechanisms. J. Mech. Mater. Struct. 2(2), 293–301 (2007)CrossRefGoogle Scholar
  14. 14.
    Guest, S.D., Fowler, P.W.: Mobility of ‘N-loops’: bodies cyclically connected by intersecting revolute hinges. Proc. R. Soc. A: Math. Phys. Eng. Sci. 466, 63–77 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Hunt, K.H.: Kinematic Geometry of Mechanisms. Clarendon Press, Oxford (1978)zbMATHGoogle Scholar
  16. 16.
    Jahn, H., Teller, E.: Stability of polyatomic molecules in degenerate electronic states. I. Orbital degeneracy. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 161(905), 220–235 (1937)CrossRefGoogle Scholar
  17. 17.
    James, G.D., Liebeck, M.W.: Representations and Characters of Groups. Cambridge University Press, Cambridge/New York (2001)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kangwai, R.D., Guest, S.D.: Detection of finite mechanisms in symmetric structures. Int. J. Solids Struct. 36(36), 5507–5527 (1999)CrossRefzbMATHGoogle Scholar
  19. 19.
    Kangwai, R.D., Guest, S.D.: Symmetry-adapted equilibrium matrices. Int. J. Solids Struct. 37, 1525–1548 (2000)CrossRefzbMATHGoogle Scholar
  20. 20.
    Kangwai, R.D., Guest, S.D., Pellegrino, S.: An introduction to the analysis of symmetric structures. Comput. Struct. 71, 671–688 (1999)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Kovács, F., Tarnai, T., Guest, S.D., Fowler, P.W.: Double-link expandohedra: a mechanical model for expansion of a virus. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460(2051), 3191–3202 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Maxwell, J.C.: On the calculation of the equilibrium and stiffness of frames. Philos. Magazine. 27, 294–299 (1864)Google Scholar
  23. 23.
    Pellegrino, S., Calladine, C.R.: Matrix analysis of statically and kinematically indeterminate frameworks. Int. J. Solids Struct. 22, 409–428 (1986)CrossRefGoogle Scholar
  24. 24.
    Quinn, C.M., McKiernan, J.G., Redmond, D.B.: Mollweide projections and symmetries on the spherical shell. J. Chem. Educ. 61, 569–571, 572–579 (1984)CrossRefGoogle Scholar
  25. 25.
    Schulze, B.: Block-diagonalized rigidity matrices of symmetric frameworks and applications. Contrib. Algebra Geom. 51(2), 427–466 (2010)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Schulze, B.: Symmetry as a sufficient condition for a finite flex. SIAM J. Discret. Math. 24(4), 1291–1312 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Schulze, B., Whiteley, W.: The orbit rigidity matrix of a symmetric framework. Discret. Comput. Geom. 46(3), 561–598 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Shirts, R.B.: Correcting two long-standing errors in point group symmetry character tables. J. Chem. Educ. 84(11), 1882–1884 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of ChemistryUniversity of SheffieldSheffieldUK
  2. 2.Department of EngineeringUniversity of CambridgeCambridgeUK
  3. 3.Fields InstituteUniversity of TorontoTorontoCanada

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